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| The Global Dynamics for a Stochastic $SIR$ Epidemic Model with Vaccination |
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| KeyWord:Vaccination, random disturbance, smooth distribution, Lyapunov function, stability |
| Author Name | Affiliation | | Shan Gao | School of mathematics and statistics, Qingdao University, Ningde Road, Qingdao, Shandong 266071, China | | Xiaoqi Sun | School of mathematics and statistics, Qingdao University, Ningde Road, Qingdao, Shandong 266071, China |
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| Abstract: |
| A new stochastic $SIR$ epidemic model with vaccination is established and its dynamical behavior is analyzed. Considering the random effects of vaccination rates and mortality in this model, it is demonstrated that the extinction and persistence of the virus is only correlated with the threshold $R^{s}_{0}$. If $R^{s}_{0}<1$, the disease dies out with probability one. And if $R^{s}_{0}>1$, the disease is stochastic persistent in the means with probability one. In addition, the existence and uniqueness of a smooth distribution are proven using the It$\hat{o}$s formula, and the sufficiency criterion is obtained using the Lyapunov function. Finally, the accuracy and efficiency of the stochastic $SIR$ epidemic model with vaccination in predicting disease transmission trends were verified through simulation. Unlike the singularity of stochastic perturbations in existing infectious disease models, the innovation of this paper is in the addition of multiple stochastic perturbations, especially distinguishing the stochastic perturbations of mortality under vaccination, which are used to study the dynamics of the model. |
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